Affine quantization of the Brans-Dicke theory: Smooth bouncingand the equivalence between the Einstein and Jordan frames

E. Frion*

PPGCosmo, CCE, Universidade Federal do Espírito Santo, Vitória, 29075-910, Espírito Santo, Braziland Institute of Cosmology and Gravitation, University of Portsmouth,

Portsmouth PO1 2EG, United Kingdom

C. R. Almeida†

COSMO—Centro Brasileiro de Pesquisas Físicas Rio de Janeiro, 22290-180, Rio de Janeiro, Braziland Max Planck Institute for the History of Science 14195 Berlin, Germany

(Received 8 October 2018; published 24 January 2019)

In this work, we present a complete analysis of the quantization of the classical Brans-Dicke theory usingthe method of affine quantization in the Hamiltonian description of the theory. The affine quantizationmethod is based on the symmetry of the phase space of the system, in this case the (positive) half-plane,which is identified with the affine group. We consider a Friedmann-Lemaître-Robertson-Walker typespacetime, and since the scale factor is always positive, the affine method seems to be more suited than thecanonical quantization for our quantum cosmology. We find the wave function of the Brans-Dicke universeand its energy spectrum. A smooth bounce is expected at the semiclassical level in the quantum phase-space portrait. We also address the problem of equivalence between the Jordan and Einstein frames.

DOI: 10.1103/PhysRevD.99.023524

I. INTRODUCTION

After the formulation of general relativity (GR), somemodified theories arose in an attempt to explain openproblems in cosmology, such as inflation and the observedaccelerated expansion. One of the oldest modifications ofGR is the Brans-Dicke theory (BDT), proposed in the early1960s by Carl H. Brans and Robert H. Dicke [1], in whichthere is a nonminimal time-dependent coupling of the long-range scalar field with geometry, that is, with gravity. TheBDT also introduces an adimensional constant ω such that,for a constant gravitational coupling, GR is recovered at thelimit ω → ∞, if the trace of the energy-momentum tensor isnot null [2–4]. Today, it is well known that, classically, theBDT is practically indistinguishable from GR, with theconstant ω estimated to be over 40 000 [5,6]. Interestingly,the Brans-Dicke scalar field arises naturally in superstringcosmology, associated with the dilaton, which couplesdirectly with the matter field [7]. The dilaton is equivalentto the graviton for a theory with dynamical gravitationalcoupling. In spite of the fact that the BDT is classically nodifferent from GR, the quantum treatment can reveal newdynamics for the primordial Universe. There are alsoclaims that the BDT cannot reproduce GR for scale-invariant matter content. In fact, in this case, it has been

shown that ω can display various effects depending on itsvalue, such as a symmetry breaking resulting in a binaryphase structure. However, for a strong coupling ω → ∞,the BDT reproduces GR only in the quantised version [8].With the assumption that quantum effects cannot be

ignored at early stages of the Universe, the quantization ofthe classical BDT in its Hamiltonian description is relevantto better understand this era. We will assume a minisuper-space, a configuration with reduced degrees of freedom forhomogeneous cosmologies, which can be understood as aprojection of the whole superspace, containing only thelargest wavelength modes of the size of the Universe [9].Minisuperspaces are considered to be toy models, sincethey reduce the superspace, that is, the observable universeon the largest scales, which have infinite degrees offreedom. However, it is still a fairly good approximationof the superspace to study certain properties. This allowsone to target specific behaviors such as the dynamics of thevolume of the Universe or to investigate the nature of theinitial singularity and the inflationary phase.We choose to explore the quantization of the BDT with

the affine quantization instead of the canonical one, sincethe domain of the variables involved (scale factor and scalarfield) is the real half-line, and its phase space can beidentified with the affine group. With this, we also avoid theoperator-ordering problem arising in the case of canonicalquantization (see, e.g., discussions in Refs. [10,11]). Theaffine quantization is also equipped with a “dequantization”

*frion.emmanuel@hotmail.fr†cralmeida00@gmail.com

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map that allows us to obtain classical expressions fromquantum operators. In the canonical quantization, theclassical measurements are obtained by the expectationvalues of the classical observables, but in the affinequantization, the classical system is recovered with possiblecorrections through this dequantization map, called quan-tum corrections or lower symbols. While being a fairlyrecent subject of interest in cosmology, the affine quanti-zation points to interesting applications, such as theremoval of divergences in nonrenormalizable theories[12,13] or the nonsingular expanding (and possibly cyclic)universes [14].This work, in which we will investigate the quantization

of the BDT applying the affine method, is a continuation ofthe analysis initiated in Ref. [15]. We present the wavefunction for a Brans-Dicke universe, and we draw itsquantum phase space. Then, it is shown that a bounce isexpected, avoiding the initial singularity. We also raise thequestion about the equivalence between the Jordan andEinstein frames. This paper is organized as follows. InSec. II, we review the classical derivation of the BDTwith aperfect fluid introduced via the Schutz formalism. InSec. III, we introduce the affine quantization method aswell as a more direct way to obtain classical estimates: thequantum phase-space portraits. In Sec. IV, we apply theaffine quantization to the BDT to obtain the Wheeler-DeWitt equation in the Jordan frame and in the Einsteinframe. Finally, we derive the semiclassical version of theHamiltonian constraint in both frames. In the last section,we present our results and discuss the dependence of theparameters on the solutions.

II. BRANS-DICKE THEORYWITH A PERFECT FLUID

The Brans-Dicke theory is characterized by the intro-duction of a scalar field nonminimally coupled to gravity,and it is described by the gravitational Lagrangian

LG ¼ ffiffiffiffiffiffi−g

p �φR − ω

φ;ρφ;ρ

φ

�: ð1Þ

The Brans-Dicke coupling parameter ω is chosen to be aconstant in this work. Let us consider a homogeneous andisotropic universe,

ds2 ¼ N2ðtÞdt2 − a2ðtÞ½dx2 þ dy2 þ dz2�; ð2Þ

where N and a are, respectively, the lapse function and thescale factor. Then, the Lagrangian (1) becomes

LG ¼ 1

N

�6½φa _a2 þ a2 _a _φ� − ωa3

_φ2

φ

�; ð3Þ

where we have already discarded the surface terms. TheLagrangian of the system is completed with a matter

component, which we will consider to be a radiative perfectfluid, defined by the equation of state p ¼ ρ=3.Let us use the Schutz formalism to introduce the perfect

fluid [16], in which the 4-velocity of a baryonic perfectfluid is described by four potentials,1 the specific enthalpyμ and the entropy s of the fluid and another two with noclear physical meaning; let us call them ϵ and θ. After someconsiderations [17,18], the matter Lagrangian becomes

LM ¼ −1

3

�3

4

�4 a3

N3ð_ϵþ θ_sÞ4e−3s: ð4Þ

Since we are interested in the quantum corrections of thissystem, we must describe the theory with the Hamiltonianformalism. To do so, let us write the Lagrangians above asfunctions of the conjugate momenta, defined by

pq ¼∂L∂ _q : ð5Þ

With this, from (4), we obtain the matter super-Hamiltonian[19]2

HM ¼ −p43ϵa−1es; ð6Þ

where pϵ ¼ −Nρ0U0a3, with ρ0 the rest mass density of thefluid and U the 4-velocity. Let us introduce the followingcanonical transformations [20]:

T ¼ −pse−sp−43

ϵ ; pT ¼ p43ϵes;

ϵ ¼ ϵ −4

3

ps

pϵ; pϵ ¼ pϵ: ð7Þ

Then, the super-Hamiltonian for the matter componentbecomes

HM ¼ −NapT: ð8Þ

The Hamiltonian for the gravitational part is given by theLegendre transformation of LG,

HG ¼ _apa þ _φpφ − LG; ð9Þwhere the conjugate momenta are

pa¼6

Nð2φa _aþa2 _φÞ; pφ ¼

6

Na2 _a−2

ω

Na3

_φ

φ: ð10Þ

Expressing the generalized velocities in terms of themomenta, we obtain

1There are six potentials in total, but they reduce to four in ahomogeneous and isotropic medium.

2The Hamiltonian defined on the minisuperspace, where thespacelike metric and nongravitational fields belong to a finite setand their conjugate momentum is identically zero.

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_a ¼ ωNð3þ 2ωÞφa

�pa

6þ φpφ

2ωa

�;

_φ ¼ Nφ

2ð3þ 2ωÞa3�apa

φ− 2pφ

�; ð11Þ

which, after some algebra, gives us

HG ¼ N3þ 2ω

�ω

12φap2a þ

1

2a2papφ −

φ

2a3p2φ

�: ð12Þ

Therefore, the Hamiltonian of the BDT is given by

H¼N

�1

ð3þ2ωÞ�

ω

12φap2aþ

1

2a2papφ−

φ

2a3p2φ

�−1

apT

�;

ð13Þ

where pT , pa, and pφ are the conjugate momenta asso-ciated with the matter component, the scale factor a, and thefield φ, respectively.The classical Hamiltonian constraint H ≈ 0 still holds

(notice that here ≈ means “weakly equal,” so that H is afirst class constraint; i.e., its Poisson brackets with otherconstraints are vanishing on the constrained space) for theBDT with a perfect fluid [21,22]. Thus, we have

ω

12φp2a þ

1

2apapφ −

φ

2a2p2φ ¼ ð3þ 2ωÞpT: ð14Þ

The quantization of this constraint results in the Wheeler-DeWitt equation. We can interpret it as a Schrödinger-likeequation and, from it, obtain the cosmological scenarios ata quantum level [23]. Now, instead of the canonicalquantization used in Ref. [23], we will introduce anotherquantization method, based on the symmetry group of thesystem’s phase space. This kind of quantization is com-pleted with a quantum phase-space portrait, which accountsfor a quantum correction to the classical trajectories ofthe theory, that we will use to analyze the BDT at earlycosmological times.

III. AFFINE QUANTIZATION

A. Mathematical background

First, let us introduce the affine quantization methodmentioned earlier. The model requires the scale factor andthe scalar field, our two dynamical variables, to be positive,with the zero value being a geometrical singularity. Thus,the phase space is a four-dimensional space which is theCartesian product of two half-planes,3

Π2þ≔ fða;paÞ× ðφ;pφÞja> 0;φ> 0;pa;pφ ∈Rg: ð15Þ

Since it is a Cartesian product, we can analyze each half-plane separately. Thus, we will present the theory behindthis method of quantization for a generic phase space andthen apply it to our specific case (for a more detailedpresentation, see e.g., Refs. [24,25]).The half-plane Πþ ≔ fðq; pÞjq > 0; p ∈ Rg with a

multiplication operation defined by

ðq;pÞðq0;p0Þ¼�qq0;

p0

qþp

�; q∈R�þ; p∈R ð16Þ

is identified with the affine group AffþðRÞ of the real line.The group acts on R as follows:

ðq; pÞ · x ¼ xqþ p; ∀x ∈ R: ð17Þ

On a physical level, one can interpret (17) as a contraction/dilation (depending on if q > 1 or q < 1) of space plus atranslation. We shall equip the half-plane with the measuredqdp, which is invariant under the left action of the affinegroup on itself [26].Rigorously, the affine quantization is a covariant integral

method, that combines the properties of symmetry from theaffine group with all the resources of integral calculus.This method makes use of coherent states [27] to constructthe quantization map, the definition of which is connectedwith the symmetry of the phase space, as we will see. First,let us explain the integral quantization method. Given agroupG and a unitary irreducible representation (UIR) of it,the quantization map transforms a classical function (ordistribution) into an operator using a bounded square-integrable operator M and a measure dν, such as

ZGMðgÞdνðgÞ ¼ I; ð18Þ

where g ∈ G, MðgÞ ¼ UðgÞMU−1ðgÞ. This is the resolu-tion of the identity for the operator M. With this, from aclassical observable fðgÞ, we obtain the correspondingoperator

Af ¼ZGMðgÞfðgÞdνðgÞ: ð19Þ

For the affine group, that is G ¼ AffþðRÞ, we havetwo nonequivalent UIR U�, plus a trivial one U0 [28,29].Let us choose U ¼ Uþ, which acts on the Hilbert spaceL2ðR�þ; dx=xαþ1Þ as

ðUðq; pÞψÞðxÞ ¼ eipxffiffiffiffiffiffiffiq−α

p ψ

�xq

�: ð20Þ

We choose the operator M such as3In the case of radiative matter, at least [23].

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M ¼ jψihψ j; ψ ∈ L2

�R�þ;

dxxαþ1

�∩ L2

�R�þ;

dxxαþ2

�:

ð21Þ

The normalized vectors ψ are arbitrarily chosen providingthe square-integrability condition (21), and they are knownas fiducial vectors. For simplicity, we will consider onlyreal fiducial vectors and will choose α ¼ −1. The action(20) of the UIR of U over fiducial vectors produces thequantum states

jq; pi ≔ Uðq; pÞjψi: ð22Þ

These states are called affine coherent states or wavelets. Itis easy to show that

ZΠþ

jq; pihq; pj dqdp2πc−1

¼ I; ð23Þ

where the constant c−1 depends on the choice of ψ and isdefined as

cγ ¼ cγðψÞ ≔Z

∞

0

jψðxÞj2 dxx2þγ : ð24Þ

Hence, the quantization maps (19) become

fðq; pÞ ↦ Af ¼ZΠþ

fðq; pÞjq; pihq; pj dqdp2πc−1

: ð25Þ

With this, one can easily verify that the quantization of theelementary functions position qβ (for any β), momentum p,and kinetic energy4 p2 yields

Aqβ ¼cβ−1c−1

Qβ; Ap ¼ −i∂∂x ¼ P;

Ap2 ¼ P2 þ cð1Þ−3c−1

Q−2; ð26Þ

with Q being the position operator defined by QfðxÞ ¼xfðxÞ and the constant cð1Þ−3 defined as

cðβÞγ ðψÞ ≔Z

∞

0

jψ ðβÞðxÞj2 dxx2þγ : ð27Þ

Notice that, in this affine quantization method, the onlydependence on the fiducial vector ψ is in the constantcoefficients of the quantum operators. Thus, the arbitrarinessof ψ does not play a fundamental role in the quantization.This is an advantage to be explored. For example, we canadjust the fiducial vectors to regain the self-adjoint character

of the operator p2 [26]. Choosing ψ such that 4cð1Þ−3 ≥ 3c−1,the kinetic operator becomes essentially self-adjoint [30],which is a desired characteristic since a Hermitian operatormust be self-adjoint. A Hermitian operator can be obtainedby imposing boundary conditions. However, there is acontinuous infinity of possible boundaries, and thus thechoice of a representation is arbitrary (this is the operator-orderingproblemof the canonical quantization). In the affinequantization, the choice of a fiducial vector can naturallyresult in an essentially self-adjoint operator, which meansthere is only one possible extension of it and, therefore, noneed to impose boundary conditions. We stress, however,that choosing fiducial vectors is not the same as choosingboundary conditions. Self-adjoint-ness is a well-knownproblem in the canonical quantization of this theory, and ithas been studied extensively in Ref. [23]. However, with theaffine quantization, we naturally recover the quantum sym-metrization of the classical product momentum position

qp ↦ Aqp ¼ c0c−1

Q Pþ P Q2

; ð28Þ

up to a constant that once again depends on the choice of thefiducial vector.

B. Quantum phase-space portraits

The construction of the affine quantization methodpresented in the previous section using coherent statesallows us to define a dequantization map, named thequantum phase-space portrait, in a very obvious way: bycalculating the expectation value of an operator withrespect to the coherent states. That is, given a quantumoperator Af, we obtain a classical function f such that

fðq; pÞ ¼ hq; pjAfjq; pi: ð29ÞIf the operator is obtained from a classical function f, assuggested in the notation, then f is a quantum correction orlower symbol of the original f [31]. It corresponds to theaverage value of fðq; pÞ with respect to the probabilitydensity distribution

ρϕðq; pÞ ¼1

2πc−1jhq; pjϕij2; ð30Þ

with jϕi ¼ jq0; p0i. We can also define the time evolutionof the distribution (30) with respect to time through aHamiltonian operator H ¼ AH, using the time evolutionoperator e−iHt. Then,

ρϕðq; p; tÞ ≔1

2πc−1jhq; pje−iHtjϕij2: ð31Þ

Thus, if you consider the operatorM ¼ ρ, the lower symbolof Af becomes [24]4Up to a factor.

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fðzÞ ¼Z

trðρðzÞρðz0ÞÞfðz0Þ d2z0

π; ð32Þ

with tr the trace. From the resolution of the identity (18),one finds trðρðzÞρðz0ÞÞ is a probability distribution of thephase space, and f is indeed an average measurement of theclassical f.From Eq. (29), using (25), the quantum correction f of a

classical function f is then

fðq;pÞ¼ 1

2πc−1

Z∞

−∞

Z∞

0

dq0dp0

qq0

Z∞

0

Z∞

0

dxdx0fðq0;p0Þ

×

�eipðx0−xÞe−ip0ðx0−xÞψ

�xq

�ψ

�x0

q

�ψ

�xq0

�ψ

�x0

q0

��:

ð33Þ

Thus, it is not necessary to find the operator Af of aclassical function f to obtain its lower symbol. One can usethe above formula (33) to do so. For example, the quantumcorrection of the classical functions qβ, p, and p2 aregiven by

qβ ¼ cβ−1c−β−2c−1

qβ; p ¼ p;

p2 ¼ p2 þ�cð1Þ−2 þ

c0cð1Þ−3

c−1

�1

q2; ð34Þ

with the constants cγ and cðβÞγ defined in (24) and (27),respectively. Notice that the corrections also depend onthe choice of specific fiducial vectors to determine theseconstants.

IV. AFFINE QUANTIZATION OF THE BDT

A. Quantization in the Jordan frame

Now that we have introduced the affine quantizationmethod and the quantum phase-space portrait coming fromit, we can apply the method to the BDT presented in Sec. II,since the variables a and φ are both positively defined.However, the Schutz variable associated to the fluid has thewhole real line as its domain, and therefore we cannotapply the affine method in it. Nevertheless, we can useanother integral quantization method based on the Weyl-Heisenberg group, which acts on the real line [32]. Here,we could also use the canonical quantization for thisvariable, since it works just fine for parameters in thewhole line, a domain that does not have any singularity and,therefore, no problems of self-adjoint-ness.5 In both cases,we have

pT ↦ PT ¼ −i∂∂T ; pT ↦ pT ¼ pT ¼ E: ð35Þ

To build the coherent states of the variables a and φ, letus name the respective fiducial vectors as ψa and ψφ, whichare a priori not the same. Then, the coherent states aregiven by

ja; pai ¼ Uajψai⇒hxja; pai ¼eipaxffiffiffi

ap ψa

�xa

�ð36Þ

jφ; pφi ¼ Uφjψφi ⇒ hyjφ; pφi ¼eipφyffiffiffi

φp ψφ

�yφ

�: ð37Þ

With this, the quantization of Eq. (14) results in thefollowing Wheeler-DeWitt equation,

�−ωλ1

1

φ∂2aþðωλ2−λ3Þ

1

φa2−λ4

1

a∂a∂φþλ5

φ

a2∂2φ

þλ61

a2∂φ

�Ψða;φ;TÞ¼−ið3þ2ωÞ∂TΨða;φ;TÞ; ð38Þ

where Ψða;φ; TÞ is the wave function. The constants λi aregiven by

λ1 ¼1

12c−1ðφÞ; λ2 ¼

1

12c−1ðφÞcð1Þ−3ðaÞc−1ðaÞ

;

λ3 ¼1

2

c−3ðaÞc−1ðaÞ

cð1Þ−2ðφÞc−1ðφÞ

; λ4 ¼1

2c−1ðaÞ;

λ5 ¼1

2

c−3ðaÞc−1ðaÞ

c0ðφÞc−1ðφÞ

; λ6 ¼1

2

c−3ðaÞc−1ðaÞ

c0ðφÞc−1ðφÞ

þ 1

4c−1ðaÞ;

ð39Þ

and we defined

cðjÞγ ðaÞ ¼Z

∞

0

½ψ ðjÞa ðxÞ�2 dx

x2þγ ;

cðjÞγ ðφÞ ¼Z

∞

0

½ψ ðjÞφ ðxÞ�2 dx

x2þγ : ð40Þ

If we choose ψa ¼ ψφ, then cðjÞγ ðaÞ ¼ cðjÞγ ðφÞ ¼ cðjÞγ . So,

let us choose a fiducial vector such that

ψa ¼ ψφ ¼ 9ffiffiffi6

p x32e−

3x2 : ð41Þ

With these vectors, we have c−2 ¼ c−1 ¼ 1, and cð1Þ−3 ¼ 3=4,which, as mentioned before, is a necessary condition for thequantized kinetic energy to be an essentially self-adjointoperator [30]. In turn, this gives the us the Wheeler-DeWittequation in the Jordan frame

5Using the Weyl-Heisenberg method can give us the advantageof introducing another constant that depends on the fiducialvector chosen in the quantization. This can be an asset used toadjust energy levels, for example.

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�−ω

12

1

φ∂2aþ

�ω

16−3

4

�1

φa2−

1

2a∂a∂φþ

φ

a2∂2φþ

5

4a2∂φ

�Ψ

¼−ið3þ2ωÞ∂TΨ: ð42Þ

From this equation, absorbing the constant 12ð3þ 2ωÞω−1

into the temporal parameter, that is, accounting it as energy,we find the Hamiltonian for the BDT in the Jordan frameto be

HJ ¼1

φ∂2a −

12

ω

�ω

16−3

4

�1

φa2þ 6

ωa∂a∂φ

−12

ω

φ

a2∂2φ −

15

ωa2∂φ: ð43Þ

It is easy to see that the Hamiltonian (43) is essentially self-adjoint for the usual measure dadφ on the Hilbert space, asexpected. One can notice that Eq. (42) is not separable. Wecan work around this problem by considering the Einsteinframe instead.

B. Conformal transformation of affine operators

The Jordan and Einstein frames are related to each otherby a conformal transformation given by gμν ¼ ϕ−1gμν,where gμν and gμν represent the metric tensors in eachframe, respectively. Thus, before analyzing the equivalencebetween these frames, let us first comment on how affineoperators change with a conformal transformation.As opposed to what happens in the canonical quantiza-

tion (see Ref. [15]), the affine operators are uniquelydefined by Eq. (25). Also, if Af is the operator obtainedfrom a classical function fðq; pÞ, with q being a positive-defined variable and p its associated momentum, thenfor a general conformal scaling factor ΩðqÞ on the domain,we have

Ω2ðqÞAf ≠ AΩ2ðqÞf: ð44Þ

Therefore, we need to be careful when we quantize modelsrelated by conformal transformations. Even if the constraintobtained from a Hamiltonian is classical, we cannot canceloverall coefficients [for instance, the factor 1=b inEq. (50)]. To illustrate this, let us give an example.Consider Ω2ðqÞ ¼ q and fðq; pÞ ¼ p. The operator AΩ2f

is given by (28), and then

AΩ2f ¼ Aqp ¼ c0c−1

QPþ PQ2

≠ QP ¼ qAp ¼ Ω2ðqÞAf:

ð45Þ

This means that, classically, it is always possible tocancel non-null coefficients; however, quantizing the con-straint in different frames can result in very differentscenarios, because of (44). In conclusion, we cannot cancel

out non-null functions before quantizing to compare thequantization of two different frames connected by a trans-formation of coordinates.

C. Quantization in the Einstein frame

Since the seminal paper of Brans and Dicke [1], we knowthat two formulations of the theory (and, in fact, for everyscalar-tensor theory) are possible. These formulations,related by a conformal transformation, are the target of along debate on which of these frames is physically relevant.Some authors claim they are equivalent classically butshould be different at the quantum level [33,34], whileothers claim that both are equivalent at classical andquantum levels [35–38]. Some also claim that the equiv-alence is broken by off-shell one-loop quantum corrections,but recovered on shell [39]. Since theoretical predictionsdepend entirely on the conformal frame we are working on,a natural question that arises is if there is a preferred frameor not, and which one is the most suitable to observations.In the Jordan frame, we found the differential equationgoverning the wave function evolution (42); however, as acrossed term appeared in the partial derivatives, finding asolution can be difficult. Let us now analyze the problem inthe Einstein frame instead.The Brans-Dicke Lagrangian, with a nonminimally

coupled scalar field, is given by (1), and by using theconformal transformation, gμν ¼ φ−1gμν, where gμν is themetric in the nonminimal coupling frame, the Lagrangianreads as

LG ¼ffiffiffiffiffiffi−g

p �R −

�ωþ 3

2

�φ;ρφ

;ρ

φ2

�; ð46Þ

which is the Lagrangian for general relativity with aminimally coupled scalar field. The Lagrangian (1) iswritten in the Jordan frame, and (46) is written in theEinstein frame. The conformal transformation is given bythe change of coordinates

N0 ¼ φ12N; b ¼ φ

12a; φ0 ¼ φ; ð47Þ

and applying these to (3), we obtain

LG ¼ 1

N0

�6b _b2 −

�ωþ 3

2

�b3� _φ0

φ0

�2�: ð48Þ

The total Hamiltonian is thus

HT ¼ N0�p2b

24b−

φ02

2ð3þ 2ωÞb3 p2φ0 −

pT

b

�; ð49Þ

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and the constraint HT ¼ 0 gives us6

p2b

24b−

φ02

2ð3þ 2ωÞb3 p2φ0 ¼ pT

b: ð50Þ

In order to quantize Eq. (50), it is necessary to know theHilbert space in the Einstein frame. From the change ofvariables (7), the measure becomes

dadφ ¼ φ0−12dbdφ0: ð51Þ

Thus, the Hilbert space for the coordinates ðb;φ0Þ isL2ðR�þ ×R�þ;φ0−1

2dbdφ0Þ. Then, according to definition(21), the fiducial vectors ψφ0 are defined on anotherHilbert space:

ψφ0 ∈ L2

�R�þ;

dx

x12

�∩ L2

�R�þ;

dx

x32

�: ð52Þ

With this measure, the operator associated with the kineticenergy is given by

Ap2 ¼ −∂2φ0 þ 1

2φ0 ∂φ0 þ�cð1Þ−5=2ðφ0Þc−1=2ðφ0Þ −

3

8

�1

φ02 ; ð53Þ

which is already self-adjoint.Now, for the coordinate b, using (25), we obtain

Ab−1p2b¼ −

1

c−1ðbÞ1

b∂2b þ

1

c−1ðbÞ1

b2∂b −

�1− cð1Þ−4ðbÞc−1ðbÞ

�1

b3:

ð54Þ

For the coordinate φ0, we get

Aφ02p2

φ0¼ −

11

8

c3=2c−1=2

þ cð1Þ−1=2

c−1=2−3

2

c3=2c−1=2

φ0∂φ0 −c3=2c−1=2

φ02∂2φ0 :

ð55Þ

Then, the quantization of Eq. (50) results in

�−ϖ∂2

bþϖ

b∂bþðλ1ϖþ λ2Þ

1

b2þ λ3b2

�φ02∂2

φ0 þ3

2φ0∂φ0

��Ψ

¼−24ϖi∂TΨ; ð56Þ

with ϖ ¼ ωþ 32, and λi are given by

λ1 ¼ cð1Þ−4ðbÞ − 1;

λ2 ¼3

4

c−4ðbÞc−1=2ðφ0Þ

�11

8c3=2ðφ0Þ − cð1Þ−1=2ðφ0Þ

�;

λ3 ¼c−4ðbÞc3=2ðφ0Þ

c−1=2ðφ0Þ : ð57Þ

On the other hand, one can change variables as in (47)directly on (38). This yields

�−�ωλ1 þ

λ42−λ54

�∂2b þ

λ5 − λ44

1

b∂b þ ðωλ2 − λ3Þ

1

b2

þ�λ52− λ4

�φ0

b∂b∂φ0 þ λ5

φ02

b2∂2φ0 þ λ6

φ0

b2∂φ0

�Ψ

¼ −ið3þ 2ωÞ∂TΨ; ð58Þ

with λi given in (39). Notice that the coefficients λi are in

terms of cðiÞλ ðaÞ and cðiÞλ ðφÞ, while the coefficients in

Eq. (56) are in terms of cðiÞλ ðbÞ and cðiÞλ ðφ0Þ. Consideringthe freedom in the choice of the fiducial vectors,7 andcomparing Eqs. (56) and (58), we conclude that there isequivalence between the Einstein and Jordan frames only if

λ52− λ4 ¼ 0 ⇒ c−3ðaÞ ¼ 2

c−1ðφÞc0ðφÞ

: ð59Þ

In a way, this result is similar to the one found in Ref. [26],in which it is concluded that the equivalence depends on thechoice of ordering factors for the canonical quantization,which are related to the coefficients of the Hamiltonianoperator. In our case, the unitary equivalence is thenobtained if we impose some constraints on the fiducialvectors:

4cð1Þ−3 ≥ 3c−1 for ψa;ψb;ψφ; and c−3ðaÞ ¼2c−1ðφÞc0ðφÞ

:

ð60Þ

Let us solve, without loss of generality, Eq. (56). Wesuppose the following separation of variables: Ψðb;φ; tÞ ≔XðbÞYðφÞPðTÞ. We obtain, for the function of time

PðTÞ ¼ A exp

�iET24

�; ð61Þ

where E=24 is the energy constant. This results in thefollowing system of partial differential equations,

6We keep the 1/b factor in order to avoid inconsistences in thequantization (see the discussion in Sec. IV. B).

7The quantization is not determined by this choice, althoughthere is an inequality constraint (4cð1Þ−3 ≥ 3c−1) in order to obtain aHermitian operator (see the discussion at the end of Sec. III. A).

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�−∂2

b þ1

b∂b þ

1

ϖ½λ1ϖ þ λ2 − λ3k2�

1

b2

�XðbÞ ¼ EXðbÞ;�

φ2∂2φ þ

3

2φ∂φ

�YðφÞ ¼ −k2YðφÞ;

ð62Þ

with k2 being a separation constant. The general solutionsare given by

XðbÞ ¼ C1bJνðffiffiffiffiE

pbÞ þ C2bYνð

ffiffiffiffiE

pbÞ; ð63Þ

YðφÞ ¼ D1φ−14ðffiffiffiffiffiffiffiffiffiffiffi1−16k2

pþ1Þ þD2φ

14ðffiffiffiffiffiffiffiffiffiffiffi1−16k2

p−1Þ; ð64Þ

with Jν and Yν the Bessel functions of first and secondkinds, respectively; C1;2 and D1;2 are integration constants,

ν ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ λ1Þϖ þ ðλ2 − λ3k2Þ

ϖ

s; ð65Þ

and k2 < 1=16. The wave function of the UniverseΨTðb;φÞ ¼ XðbÞYðφÞ must be square integrable. This isthe reason for the choice of the limit set for the separationconstant. Equation (62) is known as Euler equation, and thesolution (64) corresponds to said limit of k2. The solutionfor k2 ¼ 1=16 gives similar results; however, k2 > 1=16results in a non-square-integrable wave function. This isalso the reason why we choose a negative sign for theseparation constant. Also, since Yn blows up at the origin,we must take C2 ¼ 0. Now, let us consider the followingtransformation for the variable φ:

σ ¼ lnφ ⇒ dσ ¼ 1

φdφ: ð66Þ

With this, the solution (64) becomes8

YðσÞ ¼ D1e−σ4ðffiffiffiffiffiffiffiffiffiffiffi1−16k2

pþ1Þ þD2e

σ4ðffiffiffiffiffiffiffiffiffiffiffi1−16k2

p−1Þ: ð67Þ

For the sake of simplicity, let us consider D2 ¼ 0. Weconstruct the wave packet as

Ψ ¼ NZ

14

−14

dkbJνðffiffiffiffiE

pbÞe−σ

4ðffiffiffiffiffiffiffiffiffiffiffi1−16k2

pþ1ÞeiE24T; ð68Þ

whereN is a normalization constant. Therefore, the norm ofthe wave packets is

hΨjΨi ¼ N2

Zb0

0

Z∞

0

φ−12dbdφ

Z14

−14

Z14

−14

dkdk0b2e−σ2

× JνðffiffiffiffiE

pbÞJν0 ð

ffiffiffiffiE

pbÞeið14

ffiffiffiffiffiffiffiffiffiffiffiffij1−16k02

pj−1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffij1−16k2j

pÞσ;

ð69Þ

or, writing only in terms of σ,

hΨjΨi ¼ N2

Zb0

0

Z∞

−∞dbdσ

Z 14

−14

Z 14

−14

dkdk0b2JνðffiffiffiffiE

pbÞ

× Jν0 ðffiffiffiffiE

pbÞeið14

ffiffiffiffiffiffiffiffiffiffiffiffij1−16k02

pj−1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffij1−16k2j

pÞσ; ð70Þ

where the prime on the ν indicates νðk0Þ and we can takeb0 ¼ 1 as the value of the scale factor today. Performing theintegrals over σ and k0 gives

hΨjΨi ¼ 8πN2

Zb0

0

Z 14

−14

dbdkb2JνðffiffiffiffiE

pbÞJνð

ffiffiffiffiE

pbÞ: ð71Þ

Now, we shall consider an approximation for the limitω ≫ k2. This approximation is relevant due to our under-standing of today’s estimate of the Brans-Dicke constant ω.Notice that, in this limit, the Bessel index (65) becomes

ν ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ λ1

p, and then (71) becomes

hΨjΨi ¼ 8πN2

Zb0

0

dbb2Jν¼

ffiffiffiffiffiffiffiffi1þλ1

p ðffiffiffiffiE

pbÞJ

ν¼ffiffiffiffiffiffiffiffi1þλ1

p ðffiffiffiffiE

pbÞ:

ð72Þ

The solution is given in terms of the regularized generalizedhypergeometric function 2F3 [40] as

hΨjΨi ¼ 4ffiffiffiπ

pN2b30Γ

�νþ 1

2

�Γ�νþ 3

2

�ðb0

ffiffiffiffiE

pÞ2ν

× 2F3

�νþ 1

2;νþ 3

2;νþ 1;νþ 5

2;2νþ 1;−b20E

�:

ð73Þ

The regularized generalized hypergeometric functions aredefined as the power series

pFqða1;…; ap; b1;…; bq; zÞ

≔1

Γðb1Þ…ΓðbqÞX∞n¼0

ða1Þn…ðapÞnðb1Þn…ðbqÞn

zn

n!; ð74Þ

with the recurrence relations

ðajÞ0 ¼ 1; and

ðajÞn ¼ ajðajþ1Þðajþ2Þ…ðajþn−1Þ; for n≥ 1: ð75Þ8With this, it becomes more evident why it is only square

integrable for k2 < 1=16.

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The norm of the wave packet becomes

hΨjΨi¼Aðb0ffiffiffiffiE

pÞ2ν

X∞n¼0

ðνþ 12Þn

ðνþ1Þnð2νþ1Þnð−b20EÞn

n!; ð76Þ

where

A ¼ 4ffiffiffiπ

pN2

b30ðνþ 1Þ

Γðνþ 12Þ

Γðνþ 2ÞΓð2νþ 1Þ : ð77Þ

Then, Eq. (76) suggests that the energy spectrum isdiscrete. This means we can write hΨjΨi ¼ P

nhΨnjΨni,and the energy levels satisfy the equations

hΨ0jΨ0i ¼ Aðb0ffiffiffiffiE

pÞ2ν; ð78Þ

and for a general n ≥ 1,

hΨnjΨni ¼ Aðb0ffiffiffiffiE

pÞ2ν

X∞n¼0

ðνþ 12Þn

ðνþ 1Þnð2νþ 1Þnð−b20EÞn

n!:

ð79Þ

D. Quantum phase-space portrait of the BDT

Let us consider the formalism introduced in Sec. III. B.The constraint (14), HT ¼ 0, can be rewritten in its semi-classical version using (33) to calculate each term. For thesake of simplicity, we will keep the same letter for theenergy constant, so pT ¼ E, and hence

ω

12

1

φp2aþðωκ1−κ2Þ

1

a2φþ 1

2apapφ−κ3

φ

a2p2φ¼ð3þ2ωÞE;

ð80Þ

with the constants κi being

κ1 ¼1

12

�c0ðaÞcð1Þ−3ðaÞ

c−1ðaÞþ cð1Þ−2ðaÞ

�;

κ2 ¼1

2

c0ðaÞc−3ðaÞc−1ðaÞ

�c0ðφÞcð1Þ−3ðφÞ

c−1ðφÞþ cð1Þ−2ðφÞ

�;

κ3 ¼1

2

c0ðaÞc−3ðaÞc−1ðaÞ

c0ðφÞc−3ðφÞc−1ðφÞ

;

where cðjÞγ ðaÞ and cðjÞγ ðφÞ are

cðjÞγ ðaÞ ¼Z

∞

0

½ψ ðjÞa ðxÞ�2 dx

x2þγ ;

cðjÞγ ðφÞ ¼Z

∞

0

½ψ ðjÞφ ðxÞ�2 dx

x2þγ : ð81Þ

If we choose ψa ¼ ψφ, then cðjÞγ ðaÞ ¼ cðjÞγ ðφÞ ¼ cðjÞγ . Withthis in mind, let us choose a fiducial vector such that

ψa ¼ ψφ ¼ 9ffiffiffi6

p x32e−

3x2 : ð82Þ

With these vectors, we have c−2¼c−1¼1, and cð1Þ−3 ¼ 3=4,the latter being a necessary condition for the quantizedHamiltonian to be an essentially self-adjoint operator [30].We want this condition to hold even if we are not doingthe quantization explicitly, since the semiclassical trajecto-ries are probabilistic along the path that a quantum stateevolves. Then, Eq. (80) becomes

ω

12

1

φp2a þ

9

8ðω− 2Þ 1

a2φþ 1

2apapφ − 2

φ

a2p2φ ¼ ð3þ 2ωÞE:

ð83Þ

The expression (83) allows us to analyze the expectedbehavior of the scale factor a for the early universe, for agiven initial value of the scalar field φðt0Þ ¼ φ0 and itsmomentum at this instant pφðt0Þ ¼ pφ0.Notice that Eq. (14) is the classical Hamiltonian con-

straint in the Jordan frame. To compare the expectedbehavior of the scale factor in the Jordan frame with thatin the Einstein frame, let us calculate the quantum phase-space portrait of Eq. (50), the Hamiltonian constraint in theEinstein frame. We have

ðb−1p2bÞ ¼ p2

b

bþ cð1Þ−1ðbÞ þ c1ðbÞcð1Þ−4ðbÞ − c1ðbÞ

c−1ðbÞ1

b3; ð84Þ

ðφ02p2φ0 Þ ¼ c3=2ðφ0Þc−7=2ðφ0Þ

c−1=2ðφ0Þ φ02p2φ0 ðφ0Þ

þ�c3=2ðφ0Þcð1Þ−7=2ðφ0Þ

c−1=2ðφ0Þ þ c−3=2ðφ0Þcð1Þ−1=2ðφ0Þc−1=2ðφ0Þ

−11

8

c3=2ðφ0Þc−3=2ðφ0Þc−1=2ðφ0Þ

�: ð85Þ

Then, the quantum correction of (50) becomes

3þ 2ω

24

�p2b þ κ4

1

b2

�−

1

b2½κ5φ02p2

φ0 þ κ6� ¼ ð3þ 2ωÞE0;

ð86Þ

with E0 the energy and the constants

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κ4 ¼cð1Þ−1ðbÞ þ c1ðbÞcð1Þ−4ðbÞ − c1ðbÞ

c−1ðbÞ;

κ5 ¼1

2

c−4ðbÞc1ðbÞc−1ðbÞ

c3=2ðφ0Þc−7=2ðφ0Þc−1=2ðφ0Þ ;

κ6 ¼1

2

c−4ðbÞc1ðbÞc−1ðbÞ

c3=2ðφ0Þc−1=2ðφ0Þ

�cð1Þ−7=2ðφ0Þ

þ c−3=2ðφ0Þcð1Þ−1=2ðφ0Þc3=2ðφ0Þ −

11

8c−3=2ðφ0Þ

�:

By choosing the fiducial vectors as before, we find

3þ 2ω

24p2b þ

�1296 − 1500

ffiffiffiffiffiffi3π

p þ 864ω

64

�1

b2

−525

ffiffiffiffiffiffi3π

p

16b2φ02p2

φ0 ¼ ð3þ 2ωÞE0: ð87Þ

Equations (83) and (87) are the quantum corrections of theclassical Brans-Dicke theory described in the Jordan andEinstein frames, respectively. To understand the conse-quences of these corrections, let us build the quantum phasespace of the BDT in both these frames.

V. PHASE-SPACE PORTRAITS

As mentioned before, in this section, we present thequantum phase-space portraits coming from Eqs. (83) and(87). The aim is to understand the behavior of the scalefactor a, which is connected to the volume of the Universe,so the phase spaces shown here are with reference to thisvariable. Notice, however, that there are still other freeparameters: the scalar field φ, the energy E, and the Brans-Dicke constant ω. These parameters will be varied for the

sake of understanding their influence on the issue. Withoutloss of generality, let us consider the initial state of thescalar field to be φ0 ¼ 1.

A. Jordan frame

For the Jordan frame, let us set the energy at E0 andconstruct the phase space for a range of values of pφ

(Fig. 1). Each curve represents a value for the velocity(momentum) of the scalar field. In each plot, we have a totalof ten curves. For each curve, the lower the minimum of thescale factor is, the higher pφ is. Notice that, up until anupper value for pφ, the curves are of a smooth bouncing forthe universe, including solutions with a possible infla-tionary phase. Above a certain value of pφ, divergentcurves appear. If one assumes that this type of divergencedoes not describe a physical reality (favoring smoothness),then the scalar field must have a limit in momentum.Otherwise, this model predicts a singularity formed by anaccelerated contraction of a prior universe, reaching nullvolume as the (modulus of the) momentum goes to infinity,followed by a decelerated inflation.9

Now, we study the effect of the Brans-Dicke parameter ω(Fig. 2). In the left figure, we take ω ¼ 41 000 and see thereare more divergent lines than in the generic case consideredin Fig. 1. In the right figure, we increased ω to 4 100 000.Notice that it requires a much greater initial momentum forthe scale factor to obtain divergent solutions. Therefore, alarger ω seems to lead to a more well-behaved theory. Thisis a result of interest, since the larger ω is, the greater thecoupling between matter and the scalar field, that is, thesmaller the effects of the scalar field are. This would

FIG. 1. Quantum phase space of the scalar field in the Jordan frame, using ω ¼ 410 000 and E0 ¼ 1016. The left figure is for a range1 ≤ pφ ≤ 103, while for the right figure, the range is smaller 1 ≤ pφ ≤ 102.

9Notice that we are reading the graphics in the clockwisedirection.

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correspond to the weak-field limit we observe today.Actually, for a perfect fluid (as in our case), we recoverGR in this limit [41].The variation of the energy parameter does not change

the behavior of the solutions, but it results in a change ofscale in the phase space (Fig. 3). So, the energy candetermine the scale with which inflation happens.Up until now, we have considered the initial value of the

scalar field to be φ0 ¼ 1, but we also want to understand the

effects of the initial condition on the behavior of thesolutions (Fig. 4). Thus, in Fig. 4, we show the directinfluence of changing the value for the scalar field on thesolutions. The top row shows greater values for φ0, from 10to 104 (left to right). We notice that the greater φ0 is, themore singularities we obtain. Conversely, in the secondrow, we lower it from 0.1 to 10−4. The solutions tend tobounce instead of singularities. As expected, the results areconsistent with the study on ω.

FIG. 3. The change in the energy of the system results in a change of scale for the solutions. In the left figure, we take E ¼ 1013, and inthe right figure, we take E ¼ 10. The same values were used as before for pφ and ω: 1 ≤ pφ ≤ 103 and ω ¼ 410 000.

FIG. 2. The effect of the Brans-Dicke constant in the scalar field phase space. Once again, we use E0 ¼ 1016 and consider the range1 ≤ pφ ≤ 103. The left figure is for ω ¼ 41 000, and the right figure is for ω ¼ 4 100 000.

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FIG. 5. Quantum phase space of the scalar field in the Einstein frame, using ω ¼ 410 000 and E0 ¼ 1016. The left figure is for a range1 ≤ pφ ≤ 103, while for the right figure, the range is smaller 1 ≤ pφ ≤ 102.

FIG. 4. The top row shows the solutions for high values ofφ0: top left isφ0 ¼ 10, and top right isφ0 ¼ 104. Thebottom row is for lowvaluesof φ0: bottom left is φ0 ¼ 10−1, and bottom right is φ0 ¼ 10−4. For these, we are considering ω ¼ 410 000, E ¼ 1016, and 1 ≤ pφ ≤ 103.

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B. Einstein frame

In the Einstein frame, we have symmetric bounceswithout any inflationary epoch,10 as we see in Fig. 5. Byvarying once again ω (Fig. 6) and the energy (Fig. 7), wearrive at the same conclusions as in the Jordan frame, i.e.,

that the larger ω is, the less divergent the curves we obtainare, and varying the energy induces a scaling in the phasespace. We also show the effect of the scalar field in Fig. 8.Notice that these results are consistent with what was

found in the Jordan frame, which provides further evidencethat the frames are equivalent. Remember, though, that, inspite of choosing specific fiducial vectors, this analysis isstill qualitative, since one can always choose differentwavelets and also restore the unities (we chose c ¼ ℏ ¼ 1).For our purpose, this qualitative analysis is enough.

FIG. 6. The effect of the Brans-Dicke constant in the scalar field phase space. Once again, we take E0 ¼ 1016 and consider the range1 ≤ pφ ≤ 103. The left figure is for ω ¼ 41; 000, and the right one is for ω ¼ 4 100 000.

FIG. 7. The change in the energy of the system results in a change of scale for the solutions. In the left figure, we take E ¼ 1013, and inthe right figure, we take E ¼ 10. The same values were used as before for pφ and ω: 1 ≤ pφ ≤ 103 and ω ¼ 410 000.

10Inflation may be interpreted as a “stretching” of the solutionsinduced by the conformal transformation by going from theEinstein frame to the Jordan frame.

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VI. CONCLUSIONS

In this work, we presented the quantization of the Brans-Dicke theory using the affine covariant integral method andthe cosmological scenarios arising from it. We introducedthe classical Hamiltonian formalism of the BDT and themathematical foundations of this quantization method, inorder to familiarize the reader with the concepts used lateron. Our model is completed with the addition of a radiativematter component in form of a perfect fluid, introduced viathe Schutz formalism, which we adopted as the clock. Theaffine quantization is based on the symmetry of the phasespace of the system, and we can choose the free parameters,namely the fiducial vectors, in a way to build an essentiallyself-adjoint Hamiltonian operator. The quantization of theHamiltonian constraint results in the Wheeler-DeWittequation, from which we obtain a Schrödinger-like equa-tion (38), with the radiative matter providing the timeparameter. One expected setback of this quantization is thatit results in a nonseparable partial differential equation.

We can work around this problem by changing frames,making a conformal transformation of the coordinates.The BDT is described in the Jordan frame, and a

conformal change of coordinates transforms the BDT intoGR with a scalar field, i.e., the Einstein frame. Theequivalence between these frames is still debatable (see,e.g., Refs. [33–37]), and our results may contribute to thisdebate. In the Einstein frame, the Schrödinger-like equationis separable and becomes easier to deal with. We presentedthe classical GR with a scalar-field model corresponding tothe BDT in the Einstein frame and quantized it using theaffine method. We also performed a change of coordinatesin the already quantized Schrödinger-like equation inthe Jordan frame. Considering the freedom in the choiceof the fiducial vectors, we found an equivalent equation.However, we conclude that the Hamiltonian operator in theEinstein frame is only essentially self-adjoint if we considerdifferent fiducial vectors while quantizing the theory ineach frame, or if we change the domains (i.e., the measure)of the operators in the respective Hilbert space. In any case,

FIG. 8. The top row shows the solutions for high values of φ0: top left is φ0 ¼ 10, and top right is φ0 ¼ 104. The bottom row is for lowvalues of φ: bottom left is φ0 ¼ 10−1, and bottom right is φ0 ¼ 10−4. For these, we are considering ω ¼ 410 000, E ¼ 1016,and 1 ≤ pφ ≤ 103.

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one may argue that, because of this, there is no equivalencebetween the frames. However, the role of the fiducialvectors during the quantization is precisely to open upopportunities for adjustment, since it is based on astatistical method (jhq; pjϕij2 is interpreted as the proba-bility density distribution of the function ϕ; see, e.g.,Ref. [27]). Thus, considering different fiducial vectors indifferent frames should not invalidate the equivalencebetween them. We choose to solve the Wheeler-deWittequation obtained from the classical BDT in the Einsteinframe, in order to do a qualitative analysis, since thisequation has a relatively simple solution. From it, we wereable to conclude that the energy spectrum of theHamiltonian operator in the Einstein frame is discrete.The affine quantization method is completed with a

dequantization, known as the quantum phase-space portraitor lower symbol, that transforms the quantized operatorinto a classical function, by means of their fiducial vectorsexpectation values. This dequantization provides a quan-tum correction for classical observables, from which wecan analyze the behavior of these observables in semi-classical environments. Even if we cannot find the wavefunction of the universe in the Jordan frame, we can use thequantum phase space to compare the results with the onesfrom the quantum phase space in the Einstein frame. Thus,we find quantum corrections for the Hamiltonian constraintin both frames in Sec. IV. D and compare the results inSec. V, drawing the phase-space portrait for the scale factor,to better understand the behavior of the (volume of the)universe in earlier stages.We obtained two types of solutions in both frames:

bounces and singularities. For both types, we predict a prioruniverse. For the singular cases in the Jordan frame, there isan accelerated contraction, with a singular point where the

volume of the Universe becomes null, followed by adecelerated inflationary era. However, if we limit themomentum of the scalar field, we obtain only bouncingsolutions. Thus, we may argue that the scalar field shouldhave a limited velocity, since this discards the singularsolutions. We also analyzed the influence of other param-eters in the solutions. In the limit ω → ∞, in which weexpect to reproduce GR (for our model, at least), bouncesbecome more expected. It is interesting to see that aninflationary stage also appears for bounces in this frame. Inthe Einstein frame, however, we do not have any infla-tionary era, but similar conclusions can be drawn, with theexception that both singular and bouncing solutions aresymmetric.The use of affine quantization in cosmology is a nascent

subject, with the desirable feature of providing solutionswithout singularities for a natural range of parameters. Thisis in adequation with other results on various cosmologicalscenarios (see Refs. [42–46]) as well as a result on thequantum Belinski-Khalatnikov-Lifshitz scenario using theaffine coherent states quantization for a Bianchi IX uni-verse [47], in which they suggest that quantizing GR shouldalso lead to bouncing solutions. We intend to continue toexplore this line of research in future works.

ACKNOWLEDGMENTS

E. F. was financed in part by the Coordenaçãode Aperfeiçoamento de Pessoal de Nível Superior—Brasil—Finance Code 001 and by the Institute ofCosmology and Gravitation. E. F. and C. R. A. thankimmensely Professor Jean-Pierre Gazeau for his support,the referee for his useful comments, and also Chris Pattisonfor his diligent proofreading of this work.

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